Dean Adams, Iowa State University
15 October, 2019
\[\small\mathbf{Z}=\frac{1}{CS}\mathbf{(Y-\overline{Y})H}\]
Incorporate ‘sliding’ of semilandmarks into GPA algorithm
GPA algorithm with sliding landmarks: Procedure
Incorporate ‘sliding’ of semilandmarks into GPA algorithm
GPA algorithm with sliding landmarks: Procedure
Incorporate ‘sliding’ of semilandmarks into GPA algorithm
GPA algorithm with sliding landmarks: Procedure
Incorporate ‘sliding’ of semilandmarks into GPA algorithm
GPA algorithm with sliding landmarks: Procedure
Incorporate ‘sliding’ of semilandmarks into GPA algorithm
GPA algorithm with sliding landmarks: Procedure
Incorporate ‘sliding’ of semilandmarks into GPA algorithm
GPA algorithm with sliding landmarks: Procedure
Two analytical issues:
Slide landmarks along curve
Adjacent points on curve define sliding directions: \(\small{U}_{ij}=[(x_{j-1}-x_{j+1}),(y_{j-1}-y_{j+1})]\)
Slide landmarks along curve
Adjacent points on curve define sliding directions: \(\small{U}_{ij}=[(x_{j-1}-x_{j+1}),(y_{j-1}-y_{j+1})]\)
Assemble into tangent direction matrix
\[\tiny\mathbf{U}=\begin{bmatrix} 0 & 0 & 0 \\ u_{2x} & 0 & 0\\ 0 & u_{3x} & 0 \\ 0 & 0 & u_{4x} \\ 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ u_{2y} & 0 & 0\\ 0 & u_{3y} & 0 \\ 0 & 0 & u_{4y} \\ 0 & 0 & 0 \\ \end{bmatrix}\]
Slide landmarks along curve
Adjacent points on curve define sliding directions: \(\small{U}_{ij}=[(x_{j-1}-x_{j+1}),(y_{j-1}-y_{j+1})]\)
Assemble into tangent direction matrix
\[\tiny\mathbf{U}=\begin{bmatrix} 0 & 0 & 0 \\ u_{2x} & 0 & 0\\ 0 & u_{3x} & 0 \\ 0 & 0 & u_{4x} \\ 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ u_{2y} & 0 & 0\\ 0 & u_{3y} & 0 \\ 0 & 0 & u_{4y} \\ 0 & 0 & 0 \end{bmatrix}\]
where \(\small\mathbf{\mathcal{L}_p^{-1}}=\begin{bmatrix} \mathbf{L}_p^{-1} & 0 \\ 0 & \mathbf{L}_p^{-1} \end{bmatrix}\)
where \(\small\mathbf{\mathcal{L}_p^{-1}}=\begin{bmatrix} \mathbf{L}_p^{-1} & 0 \\ 0 & \mathbf{L}_p^{-1} \end{bmatrix}\)
\[\tiny\mathbf{U}=\begin{bmatrix} 0 & 0 & 0 & 0 & 0 \\ 0 & u_{2x} & 0 & 0& 0 \\ 0 & 0 & u_{3x} & 0 & 0 \\ 0 & 0 & 0 & u_{4x} & 0 \\ 0 & 0 & 0 & 0 & 0 \\ \hline 0 & 0 & 0 & 0 & 0 \\ 0 & u_{2y} & 0 & 0 & 0\\ 0 & 0 & u_{3y} & 0 & 0 \\ 0 & 0 & 0 & u_{4y} & 0 \\ 0 & 0 & 0 & 0 & 0 \end{bmatrix}\]
geomorph uses very clever algebra to eliminate redundancies in computations to speed this process up considerablyCranial profiles in Homo
External differences in frontal bones of archaic and modern humans are well-known
Archaic humans have ‘frontal flattening’ while modern human crania are more vertically rounded
Common anthropological interpretation: more-rounded crania in modern humans due to enlargement of frontal lobes of brain
Profiles of internal and external frontal bones were digitized from 5 mid-Pleistocene and Neanderthal crania, and 16 modern humans and compared
\[\tiny\mathbf{U}=\begin{bmatrix} 0 & 0 & 0 \\ u_{2x} & 0 & 0\\ 0 & u_{3x} & 0 \\ 0 & 0 & u_{4x} \\ 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ u_{2y} & 0 & 0\\ 0 & u_{3y} & 0 \\ 0 & 0 & u_{4y} \\ 0 & 0 & 0 \\ \hline 0 & 0 & 0 \\ u_{2z} & 0 & 0\\ 0 & u_{3z} & 0 \\ 0 & 0 & u_{4z} \\ 0 & 0 & 0 \end{bmatrix}\]
\[\tiny\mathbf{T}=(\mathbf{U^T\mathcal{L}_p^{-1}U})^{-1}\mathbf{\mathcal{L}_p^{-1}U^T}(\mathbf{Y^0-Y_{ref}})\] \[\tiny\mathbf{\mathcal{L}_p^{-1}}=\begin{bmatrix} \mathbf{L}_p^{-1} & 0 & 0 \\ 0 & \mathbf{L}_p^{-1} & 0 \\ 0 & 0 & \mathbf{L}_p^{-1} \end{bmatrix}\]
\[\tiny\mathbf{T}=\mathbf{U^T}(\mathbf{Y^0-Y_{ref}})\]
\[\tiny\mathbf{U}= \left[ \begin{array}{ccc|ccc} 0 & 0 & 0 & 0 & 0 & 0 \\ u_{2x} & 0 & 0 & w_{2x} & 0 & 0 \\ 0 & u_{3x} & 0 & 0 & w_{3x} & 0 \\ 0 & 0 & u_{4x} & 0 & 0 & w_{4x} \\ 0 & 0 & 0 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 & 0 & 0\\ u_{2y} & 0 & 0 & w_{2y} & 0 & 0 \\ 0 & u_{3y} & 0 & 0 & w_{3y} & 0 \\ 0 & 0 & u_{4y} & 0 & 0 & w_{4y} \\ 0 & 0 & 0 & 0 & 0 & 0\\ \hline 0 & 0 & 0 & 0 & 0 & 0\\ u_{2z} & 0 & 0 & w_{2z} & 0 & 0\\ 0 & u_{3z} & 0 & 0 & w_{3z} & 0 \\ 0 & 0 & u_{4z} & 0 & 0 & w_{4z} \\ 0 & 0 & 0 & 0 & 0 & 0 \\ \end{array} \right]\]
geomorph uses clever algorithms to speed this up! [ask Mike!])